Characterizing groupoid C*-algebras of non-Hausdorff étale groupoids

Abstract

Given a not-necessarily Hausdorff, topologically free, twisted étale groupoid $(G,L)$, we consider its "essential groupoid C*-algebra", denoted $C_{ess}^{\ast }(G,L)$, obtained by completing $C_c(G,L)$ with the smallest among all C*-seminorms coinciding with the uniform norm on $C_c(G^0)$. The inclusion of C*-algebras $(C_0(G^0),C_{ess}^{\ast }(G,L))$ is then proven to satisfy a list of properties characterizing it as what we call a "weak Cartan inclusion". We then prove that every weak Cartan inclusion $(A,B)$, with $B$ separable, is modeled by a topologically free, twisted étale groupoid, as above. In our second main result we give a necessary and sufficient condition for an inclusion of C*-algebras $(A,B)$ to be modeled by a twisted étale groupoid based on the notion of "canonical states". A simplicity criterion for $C_{ess}^{\ast }(G,L)$ is proven and many examples are provided.

Version accepted for publication

References in corpus (4)

• Inverse semigroup actions on groupoids
• A non-amenable groupoid whose maximal and reduced $C^{\ast }$-algebras are the same
• Noncommutative Cartan sub-algebras of C*-algebras
• Cartan Triples

Cited by in corpus (1)

• Simplicity of algebras and $C^{\ast }$-algebras of self-similar groupoids